0
votes
0answers
18 views

is it all right my pf?

PB: Give a proof that the image of a circle under a linear transformation is a circle. (Let $z$ be a $z=z_{0}+Re^{it}$, $t$ is a angle.) I tried it. Can you check my pf? (is it all right?) My Pf) ...
0
votes
1answer
30 views

Real linear tranformation

When do we say that a transformation $T$ which takes the complex number field onto itself is real-linear? I need to know it for my homework but I can't seem to find the definition anywhere.
-1
votes
1answer
80 views

The image under mapping $w=(z+i)/(z-i)$, of the third quadrant?

The title says it all. I am not sure how to approach this problem. The only related problems i have done is mapping a (unbounded)line /circle to a line/circle. Regards Exatic
0
votes
1answer
46 views

Mapping behavior of imaginary axis via $v=\frac{z-a}{z+a}$

I would like to know what the bilinear transform $v=\frac{z-a}{z+a}$ does to the imaginary axis, where $a$ is a real number. I substituted $z=yi$ and calculated $|v|$ giving me $|v| =1$. Is this ...
0
votes
1answer
39 views

inverse transform of $Z(\omega) =\frac{a}{\alpha-i\omega}$

I am stuck at calculating the inverse transorm of $Z(\omega) =\frac{a}{\alpha-i\omega}$. Can someone help me please? thanks
0
votes
3answers
424 views

A Möbius transformation maps circles and lines to circles and lines. What exactly does that mean?

The title pretty much says it all. I am also looking for a concrete example if possible. I have looked at the proof, but I'm not exactly sure what it means because I am kind of confused on what the ...
1
vote
1answer
70 views

Möbius transformation question

Möbius transformation copies the annulus $\{z:r<|z|<1\}$ to the domain between $\{z:|z-1/4|=1/4\}$ and $\{z:|z|=1\}$ Please help me to find what is $r$.
0
votes
1answer
69 views

What is the image of $D=\{z:0<\operatorname{Re}z<\pi\}\setminus\{\pi/2\}$ under $f(z)=\tan z$?

What would be the image of the domain $D = \{z:0<\operatorname{Re}z<\pi\} \setminus \{\pi/2\}$ under $f(z) = \tan z$? I havn't met with tan(z) transformation so I don't really know how to ...
1
vote
1answer
81 views

Harmonic Function Transformation Help

Consider the harmonic function $$u(x,y)=1-y+\frac{x}{x^2+y^2}$$ on the upper half plane $y>0$. What is the corresponding harmonic function on the first quadrant $x>0$, $y>0$, under the ...
0
votes
1answer
75 views

image of a circle under conformal trasformation

Consider a circle: $C_R=\{w=(x,y): |w|^2=x^2+y^2=R^2\}$ Prove that $A(C_R)$ remains a circle if $A$ is either a conformal or an anticonformal matrix. My attempt: I defined the complex number $z:=x ...
2
votes
1answer
76 views

Finding a hyperbolic isometry that fixes the point $x = 2$ and $x = 17$

I know that a Möbius transformation is hyperbolic if the trace is $> 2$ which is $a + d$. But I'm not sure of the next steps involved to arrive at the answer.
2
votes
2answers
139 views

Find an orientation preserving isometry $f (z) = \frac{az+b}{cz+d}$ such that $f (i) = 17 + 3i$

This is probably a very simple questions but I am not clear on Möbius transformations and how to solve this problem. I'd appreciate if somebody can point me towards a method to do these sort of ...
0
votes
0answers
120 views

Proof of identity of magnitude of rational function in z-domain

In the set of rational polynomial functions $H(z)$ of a complex number $z$, there exist functions whose magnitude $|H(z)|^2$ is a constant $C$, but whose denominator and numerator are not constants. ...
1
vote
1answer
88 views

Region of convergence of Z-Transform connected area?

Shouldn't the Region of Convergence of the Z transform be a connected area ? In Oppenheim solution manual, I've found this answer of a question that asks to determine the different forms of the ...
3
votes
1answer
338 views

Find Möbius transformation that send Re(z)=Im(z) to a circle and the real axis to itself

Problem 3.3.7d in Complex Variables, 2nd edition, by Stephen D. Fisher. Find a linear fractional transformation $T$ that maps the real axis onto itself and the line $y=x$ onto the circle ...
3
votes
1answer
371 views

Möbius Transformation

Hey I am doing a basic undergraduate course in complex analysis and need some help on Möbius transformations. When determining the Möbius transformation does it really matter what 3 points I'm ...
0
votes
2answers
994 views

Any linear fractional transformation transforming the real axis to itself can be written in terms of reals?

I'm trying to teach myself complex analysis, and was reading about linear transformations. I would like to understand why any linear fractional transformation which transforms the real axis into ...